Results in Mathematics

, Volume 38, Issue 1–2, pp 152–165 | Cite as

The sets of abstract trigonometric series induced by Lp



The constructive character of Lp and the rate-summability provide starting-points for the present investigation. This paper considers the constructive spaces L p and \(\mathcal{L}_{T\lambda }^p\). The constructive character of these spaces is determined by the rate λ. By appropriate λ’s the L p forms the subspaces of Lp and the \(\mathcal{L}_{T\lambda }^p\) forms the abstract sets of Fourier-Schwartz series. The constructive spaces preserve (in generalized form) the structural properties of Lp. It is used to develop the G. Goes method of complementary spaces and to consider the multipliers between different constructive spaces.

1991 Mathematics Subject Classification

40A30 42A24 42A45 

Key words and phrases

trigonometric series summability multipliers 


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  1. [1]
    Butzer P.L., Scherer K., On the fundamental approximation theorems of D. Jackson, S. N. Berstein and theorems of M. Zamansky and S. B. Steĉkin. Aequationes math., 1969, 3, 170–185.MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Edwards R. E., Fourier Series. A modern introduction. Vol.I and II., Holt, Rinehart and Winston, New York, etc., 1967.Google Scholar
  3. [3]
    Goes G., BK-Räume und Matrixtransformationen für Fourierkoeffizienten. Math. Z., 1959, 70, 345–371.MathSciNetMATHCrossRefGoogle Scholar
  4. [4]
    Goes G., Komplementräre Fourierkoeffizientenräume und Multiplikatoren. Math. Ann., 1959, 137, 371–384.MathSciNetMATHCrossRefGoogle Scholar
  5. [5]
    Kangro G., Summability factors for the series λ-bounded by the methods of Riez and Cesàro. Tartu Ülik. Toimetised, 1971, 277, 136–154. (in Russian)Google Scholar
  6. [6]
    Kralik D., Elementary method of the theory of series in approximation theory. Math. Lapok., 1980,28, 27–33.MathSciNetMATHGoogle Scholar
  7. [7]
    Sikk J., Complementary spaces with rapidity of Fourier coefficients. Tartu Ülik. Toimetised, 1975, 355, 222–235. (in Russian)MathSciNetGoogle Scholar
  8. [8]
    Sikk J., Matrix mappings for rate-spaces and K-multipliers in the theory summability. Acta et Comm. Univ. Tartuensis, 1989, 846, 118–128.MathSciNetGoogle Scholar
  9. [9]
    Sikk J., The rate-spaces m(λ), c(λ), c0(λ) and ℓp(λ) of sequences. Acta et Comm. Univ. Tartuensis, 1994, 970, 87–96.MathSciNetGoogle Scholar
  10. [10]
    Tõnnov M., Summability factors, Fourier coefficients and multiplicators. Tartu Ülik. Toimetised, 1966, 192, 82–97. (in Russian)Google Scholar

Copyright information

© Birkhäuser Verlag, Basel 2000

Authors and Affiliations

  1. 1.Department of MathematicsEstonian Agricultural UniversityTartuEstonia

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